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(3^n-1)/6^n

  • How to use it?

  • Sum of series:
  • (3^n-1)/6^n (3^n-1)/6^n
  • 1/3 1/3
  • 1/4n^2-1 1/4n^2-1
  • x^(n-1)/factorial(n-1)

  • Identical expressions

  • (three ^n- one)/ six ^n
  • (3 to the power of n minus 1) divide by 6 to the power of n
  • (three to the power of n minus one) divide by six to the power of n
  • (3n-1)/6n
  • 3n-1/6n
  • 3^n-1/6^n
  • (3^n-1) divide by 6^n

  • Similar expressions

  • (3^n+1)/6^n
Sum of series/ (3^n-1)/6^n

Sum of series (3^n-1)/6^n


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \     n    
  \   3  - 1
   )  ------
  /      n  
 /      6   
/___,       
n = 0
$$\sum_{n=0}^{\infty} \frac{3^{n} - 1}{6^{n}}$$ 
Sum((3^n - 1)/6^n, (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{3^{n} - 1}{6^{n}}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = 3^{n} - 1$$
and
$$x_{0} = -6$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-6 + \lim_{n \to \infty} \left|{\frac{3^{n} - 1}{3^{n + 1} - 1}}\right|\right)$$
Let's take the limit
we find
False

False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
4/5
$$\frac{4}{5}$$ 
4/5
Numerical answer [src] 
0.800000000000000000000000000000
0.800000000000000000000000000000
The graph
Sum of series (3^n-1)/6^n

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